Thermal Physics Kittel chapter 6 -- Entropy of mixing problem

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Homework Statement
Suppose that a system of N atoms of type A is placed
in diffusive contact with a system of N atoms of type B at the same temperature
and volume. Show that after diffusive equilibrium is reached the total entropy
is increased by 2N log 2. The entropy increase 2N log 2 is known as the entropy
of mixing. If the atoms are identical (A = B), show that there is no increase in
entropy when diffusive contact is established. The difference in the results has
been called the Gibbs paradox.
Relevant Equations
sigma = log(g), mu = tau*log(N/V*n_Q), sigma = N[log(V*n_Q/N)+5/2]
I've been working on this problem for the past 3 days. I have other papers with different ways of tackling the problem. However, I just cannot get to the answer (change in entropy = 2Nlog(2)).
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Welcome to PF.

Your attached picture of your work is very light and hard to read. Can you upload a better image please?

Also, I will send you a message with tips for posting math equations at PF using LaTeX. That is a much better way to show your work in the future here. :smile:
 

Thread 'Please tell me where I am going wrong in this integral'

##|\Psi|^2=\frac{1}{\sqrt{\pi b^2}}\exp(\frac{-(x-x_0)^2}{b^2}).## ##\braket{x}=\frac{1}{\sqrt{\pi b^2}}\int_{-\infty}^{\infty}dx\,x\exp(-\frac{(x-x_0)^2}{b^2}).## ##y=x-x_0 \quad x=y+x_0 \quad dy=dx.## The boundaries remain infinite, I believe. ##\frac{1}{\sqrt{\pi b^2}}\int_{-\infty}^{\infty}dy(y+x_0)\exp(\frac{-y^2}{b^2}).## ##\frac{2}{\sqrt{\pi b^2}}\int_0^{\infty}dy\,y\exp(\frac{-y^2}{b^2})+\frac{2x_0}{\sqrt{\pi b^2}}\int_0^{\infty}dy\,\exp(-\frac{y^2}{b^2}).## I then resolved the two...
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Thread 'Number of quantum accessible states of a particle given T, N?'

It's given a gas of particles all identical which has T fixed and spin S. Let's ##g(\epsilon)## the density of orbital states and ##g(\epsilon) = g_0## for ##\forall \epsilon \in [\epsilon_0, \epsilon_1]##, zero otherwise. How to compute the number of accessible quantum states of one particle? This is my attempt, and I suspect that is not good. Let S=0 and then bosons in a system. Simply, if we have the density of orbitals we have to integrate ##g(\epsilon)## and we have...
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